You will hate yourself if you do stack up number two ;) Maybe that's harsh but it's a going to be a PITA reworking a board with all internal signals. Don't be afraid of vias either.
Let's address some of your questions:
1.Signal layers are adjacent to ground planes.
Stop thinking about ground planes, and think more about reference planes. A signal running over a reference plane, whose voltage happens to be at VCC will still return over that reference plane. So the argument that somehow having your signal run over GND and not VCC is better is basically invalid.
2.Signal layers are tightly coupled (close) to their adjacent planes.
See number one I think the misunderstanding about only GND planes offering a return path leads to this misconception. What you want to do is keep your signals close to their reference planes, and at a constant correct impedance...
3.The ground planes can act as shields for the inner signal layers. (I think this requires stitching ??)
Yeah you could try to make a cage like this I guess, for your board you'll get better results keeping your trace to plane height as low as possible.
4.Multiple ground planes lower the ground (reference plane) impedance of the board and reduce the common-mode radiation. (don't really understand this one)
I think you've taken this to mean the more gnd planes I have the better, which is not really the case. This sounds like a broken rule of thumb to me.
My recommendation for your board based only on what you've told me is to do the following:
(thin maybe 4-5mil FR4)
(main FR-4 thickness, maybe 52 mil more or less depending on your final thickness)
(thin maybe 4-5mil FR4)
Make sure you decouple properly.
Then if you really want to get into this go to amazon and buy either Dr Johnson's Highspeed digital design a handbook of black magic, or maybe Eric Bogatin's Signal and Power integrity Simplified. Read it love, live it :) Their websites have great information as well.
I suggest you combine the two methods. Drill a hole near the traces and thread a thin bare wire through the board, lay it along the exposed traces by a few mm and solder it on both sides.
If you don't have such a wire handy, just strip a piece of stranded wire and use one of the strands
Nobody will notice unless they look very carefully, and it will be reliable.
As Peter Bennett said, the air gap layer is probably a Gerber containing areas to be milled out of the layers, possibly the top and bottom prepreg, leaving the core intact. Since there are only 4 copper layers, this would likely leave open cavities on the top and bottom with copper potentially exposed on the power/ground layers.
This could be used to recess components into the PCB.
In some cases, components are completely embedded into the PCB.
I believe this process typically would have the (in this case) core run through a pick and place machine, soldered, cleaned and then laminated and the holes plated through with the top and bottom prepreg.
Here is an example of a stackup with completely embedded components from Altium: